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Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY Lesson 17: The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to Be Similar Student Outcomes ๏ง Students prove the side-angle-side criterion for two triangles to be similar and use it to solve triangle problems. ๏ง Students prove the side-side-side criterion for two triangles to be similar and use it to solve triangle problems. Lesson Notes At this point, students know that two triangles can be considered similar if they have two pairs of corresponding equal angles. In this lesson, students learn the other conditions that can be used to deem two triangles similar, specifically the SAS and SSS criteria. Classwork Opening Exercise (3 minutes) Opening Exercise a. Choose three lengths that represent the sides of a triangle. Draw the triangle with your chosen lengths using construction tools. Answers will vary. Sample response: ๐ ๐๐ฆ, ๐ ๐๐ฆ, and ๐ ๐๐ฆ. b. Multiply each length in your original triangle by ๐ to get three corresponding lengths of sides for a second triangle. Draw your second triangle using construction tools. Answers will be twice the lengths given in part (a). Sample response: ๐๐ ๐๐ฆ, ๐๐ ๐๐ฆ, and ๐๐ ๐๐ฆ. c. Do your constructed triangles appear to be similar? Explain your answer. The triangles appear to be similar. Their corresponding sides are given as having lengths in the ratio ๐: ๐, and the corresponding angles appear to be equal in measure. (This can be verified using either a protractor or patty paper.) MP.3 d. Do you think that the triangles can be shown to be similar without knowing the angle measures? Answers will vary. ๏ง In Lesson 16, we discovered that if two triangles have two pairs of angles with equal measures, it is not necessary to check all 6 conditions (3 sides and 3 angles) to show that the two triangles are similar. We called it the AA criterion. In this lesson, we will use other combinations of conditions to conclude that two triangles are similar. Lesson 17: The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 255 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY Exploratory Challenge 1/Exercises 1โ2 (8 minutes) This challenge gives students the opportunity to discover the SAS criterion for similar triangles. The Discussion that follows solidifies this fact. Consider splitting the class into two groups so that one group works on Exercise 1 and the other group completes Exercise 2. Then, have students share their work with the whole class. Exploratory Challenge 1/Exercises 1โ2 1. Examine the figure, and answer the questions to determine whether or not the triangles shown are similar. Scaffolding: ๏ง It may be helpful to ask students how many triangles they see in Figure 1, and then have them identify each of the triangles. a. What information is given about the triangles in Figure 1? We are given that โ ๐จ is common to both โณ ๐จ๐ฉ๐ช and โณ ๐จ๐ฉโฒ ๐ชโฒ . We are also given information about some of the side lengths. b. How can the information provided be used to determine whether โณ ๐จ๐ฉ๐ช is similar to โณ ๐จ๐ฉโฒ ๐ชโฒ ? ๏ง Have students create triangles that meet specific criteria. Given a starting triangle with lengths 3, 6, 7, generate similar triangles by multiplying the sides of the โstartingโ triangle by a single factor, such as 1, 1.5, 2, 3, and so on. We know that similar triangles will have ratios of corresponding sides that are proportional; therefore, we can use the side lengths to check for proportionality. c. Compare the corresponding side lengths of โณ ๐จ๐ฉ๐ช and โณ ๐จ๐ฉโฒ ๐ชโฒ. What do you notice? ๐ ๐ = ๐๐ ๐. ๐๐ ๐๐ = ๐๐ The cross-products are equal; therefore, the side lengths are proportional. d. Based on your work in parts (a)โ(c), draw a conclusion about the relationship between โณ ๐จ๐ฉ๐ช and โณ ๐จ๐ฉโฒ ๐ชโฒ . Explain your reasoning. The triangles are similar. By the triangle side splitter theorem, I know that when the sides of a triangle are ฬ ฬ ฬ ฬ โฅ ฬ ฬ ฬ ฬ ฬ ฬ cut proportionally, then ๐ฉ๐ช ๐ฉโฒ ๐ชโฒ . Then, I can conclude that the triangles are similar because they have two pairs of corresponding angles that are equal. Lesson 17: The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 256 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY 2. Examine the figure, and answer the questions to determine whether or not the triangles shown are similar. Scaffolding: It may be helpful to ask students how many triangles they see in Figure 2; then, have them identify each of the triangles. a. What information is given about the triangles in Figure 2? We are given that โ ๐ท is common to both โณ ๐ท๐ธ๐น and โณ ๐ท๐ธโฒ๐นโฒ. We are also given information about some of the side lengths. b. How can the information provided be used to determine whether โณ ๐ท๐ธ๐น is similar to โณ ๐ท๐ธโฒ๐นโฒ? We know that similar triangles will have ratios of corresponding sides that are proportional; therefore, we can use the side lengths to check for proportionality. c. Compare the corresponding side lengths of โณ ๐ท๐ธ๐น and โณ ๐ท๐ธโฒ๐นโฒ. What do you notice? ๐ ๐ โ ๐๐ ๐ ๐๐ โ ๐๐ The side lengths are not proportional. d. Based on your work in parts (a)โ(c), draw a conclusion about the relationship between โณ ๐ท๐ธ๐น and โณ ๐ท๐ธโฒ๐นโฒ. Explain your reasoning. The triangles are not similar. The side lengths are not proportional, which is what I would expect in similar triangles. I know that the triangles have one common angle, but I cannot determine from the information given whether there is another pair of equal angles. Therefore, I conclude that the triangles are not similar. Discussion (5 minutes) Have students reference their work in part (d) of Exercises 1 and 2 while leading the discussion below. ๏ง Which figure contained triangles that are similar? How do you know? ๏บ ๏ง Figure 1 has the similar triangles. I know because the triangles have side lengths that are proportional. Since the side lengths are split proportionally by segment ๐ต๐ถ, then ๐ต๐ถ โฅ ๐ตโฒ ๐ถ โฒ , and the triangles are similar by the AA criterion. The same cannot be said about Figure 2. The triangles in Figure 1 are similar. Take note of the information that is given about the triangles in the diagrams, not what you are able to deduce about the angles. ๏บ The diagram shows information related to the side lengths and the angle between the sides. Lesson 17: The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 257 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY Once the information contained in the above two bullets is clear to students, explain the side-angle-side criterion for triangle similarity below. THEOREM: The side-angle-side criterion for two triangles to be similar is as follows: Given two triangles ๐ด๐ต๐ถ and ๐ดโฒ๐ตโฒ๐ถโฒ so that ๐ดโฒ ๐ต โฒ ๐ด๐ต = ๐ดโฒ ๐ถ โฒ ๐ด๐ถ and ๐โ ๐ด = ๐โ ๐ดโฒ , the triangles are similar; โณ ๐ด๐ต๐ถ ~ โณ ๐ดโฒ ๐ตโฒ ๐ถ โฒ . That is, two triangles are similar if they have one pair of corresponding angles that are congruent and the sides adjacent to that angle are proportional. ๏ง The proof of this theorem is simply to take any dilation with a scale factor of ๐ = ๐ดโฒ ๐ต โฒ ๐ด๐ต = ๐ดโฒ ๐ถ โฒ ๐ด๐ถ . This dilation maps โณ ๐ด๐ต๐ถ to a triangle that is congruent to โณ ๐ดโฒ๐ตโฒ๐ถโฒ by the side-angle-side congruence criterion. ๏ง We refer to the angle between the two sides as the included angle. Or we can say the sides are adjacent to the given angle. When the side lengths adjacent to the angle are in proportion, then we can conclude that the triangles are similar by the side-angle-side criterion. The question below requires students to apply their new knowledge of the SAS criterion to the triangles in Figure 2. Students should have concluded that they were not similar in part (d) of Exercise 2. The question below pushes students to apply the SAS and AA theorems related to triangle similarity to show definitively that the triangles in Figure 2 are not similar. ๏ง Does Figure 2 have side lengths that are proportional? What can you conclude about the triangles in Figure 2? ๏บ ๏ง No, the side lengths are not proportional, so we cannot use the SAS criterion for similarity to say that the triangles are similar. If the side lengths were proportional, we could conclude that the lines containing ฬ ฬ ฬ ฬ ๐๐ and ฬ ฬ ฬ ฬ ฬ ฬ ๐โฒ ๐โฒ are parallel, but the side lengths are not proportional; therefore, the lines containing ฬ ฬ ฬ ฬ ๐๐ and ฬ ฬ ฬ ฬ ฬ ฬ ๐โฒ ๐โฒ are not parallel. This then means that the corresponding angles are not equal, and the AA criterion cannot be used to say that the triangles are similar. We can use the SAS criterion to determine if a pair of triangles are similar. Since the side lengths in Figure 2 were not proportional, we can conclude that the triangles are not similar. Exploratory Challenge 2/Exercises 3โ4 (8 minutes) This challenge gives students the opportunity to discover the SSS criterion for similar triangles. The Discussion that follows solidifies this fact. Consider splitting the class into two groups so that one group works on Exercise 3 and the other group completes Exercise 4. Then, have students share their work with the whole class. Exploratory Challenge 2/Exercises 3โ4 3. Examine the figure, and answer the questions to determine whether or not the triangles shown are similar. Lesson 17: The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 258 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY a. What information is given about the triangles in Figure 3? We are only given information related to the side lengths of the triangles. b. How can the information provided be used to determine whether โณ ๐จ๐ฉ๐ช is similar to โณ ๐จโฒ๐ฉโฒ ๐ชโฒ ? We know that similar triangles will have ratios of corresponding sides that are proportional; therefore, we can use the side lengths to check for proportionality. c. Compare the corresponding side lengths of โณ ๐จ๐ฉ๐ช and โณ ๐จโฒ๐ฉโฒ ๐ชโฒ . What do you notice? ๐. ๐๐ ๐. ๐ ๐. ๐ ๐ = = = ๐. ๐๐ ๐ ๐. ๐ ๐ The side lengths are proportional. d. Based on your work in parts (a)โ(c), make a conjecture about the relationship between โณ ๐จ๐ฉ๐ช and โณ ๐จโฒ๐ฉโฒ ๐ชโฒ . Explain your reasoning. I think that the triangles are similar. The side lengths are proportional, which is what I would expect in similar triangles. 4. Examine the figure, and answer the questions to determine whether or not the triangles shown are similar. a. What information is given about the triangles in Figure 4? We are only given information related to the side lengths of the triangles. b. How can the information provided be used to determine whether โณ ๐จ๐ฉ๐ช is similar to โณ ๐จโฒ๐ฉโฒ ๐ชโฒ ? We know that similar triangles will have ratios of corresponding sides that are proportional; therefore, we can use the side lengths to check for proportionality. Lesson 17: The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 259 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY c. Compare the corresponding side lengths of โณ ๐จ๐ฉ๐ช and โณ ๐จโฒ๐ฉโฒ ๐ชโฒ . What do you notice? ๐. ๐ ๐. ๐๐ ๐. ๐๐ โ โ ๐. ๐๐ ๐. ๐๐ ๐. ๐๐ The side lengths are not proportional. d. Based on your work in parts (a)โ(c), make a conjecture about the relationship between โณ ๐จ๐ฉ๐ช and โณ ๐จโฒ๐ฉโฒ ๐ชโฒ . Explain your reasoning. I think that the triangles are not similar. I would expect the side lengths to be proportional if the triangles were similar. Discussion (5 minutes) ๏ง MP.8 Which figure contains triangles that are similar? What made you think they are similar? ๏บ ๏ง The triangles in Figure 3 are similar. Take note of the information given about the triangles in the diagrams. ๏บ ๏ง Figure 3 has the similar triangles. I think Figure 3 has the similar triangles because all three pairs of corresponding sides are in proportion. That is not the case for Figure 4. The diagram shows information related only to the side lengths of each of the triangles. THEOREM: The side-side-side criterion for two triangles to be similar is as follows. Given two triangles ๐ด๐ต๐ถ and ๐ดโฒ๐ตโฒ๐ถโฒ so that โฒ โฒ โฒ ๐ดโฒ ๐ต โฒ ๐ด๐ต = ๐ตโฒ ๐ถ โฒ ๐ต๐ถ = ๐ดโฒ ๐ถ โฒ ๐ด๐ถ , the triangles are similar; โณ ๐ด๐ต๐ถ ~ โณ ๐ด ๐ต ๐ถ . In other words, two triangles are similar if their corresponding sides are proportional. ๏ง What would the scale factor, ๐, need to be to show that these triangles are similar? Explain. ๏บ A scale factor of ๐ = ๐ดโฒ ๐ต โฒ ๐ด๐ต or ๐ = ๐ตโฒ ๐ถ โฒ ๐ต๐ถ or ๐ = ๐ดโฒ ๐ถ โฒ ๐ด๐ถ would show that these triangles are similar. Dilating by one of those scale factors guarantees that the corresponding sides of the triangles will be proportional. ๏ง Then, a dilation by scale factor ๐ = ๐ดโฒ ๐ต โฒ ๐ด๐ต = ๐ตโฒ ๐ถ โฒ ๐ต๐ถ = โณ ๐ดโฒ๐ตโฒ๐ถโฒ by the side-side-side congruence criterion. ๐ดโฒ ๐ถ โฒ ๐ด๐ถ maps โณ ๐ด๐ต๐ถ to a triangle that is congruent to ๏ง When all three pairs of corresponding sides are in proportion, we can conclude that the triangles are similar by the side-side-side criterion. ๏ง Does Figure 4 have side lengths that are proportional? What can you conclude about the triangles in Figure 4? ๏บ ๏ง No, the side lengths are not proportional; therefore, the triangles are not similar. We can use the SSS criterion to determine if two triangles are similar. Since the side lengths in Figure 4 are not proportional, we can conclude that the triangles are not similar. Exercises 5โ10 (8 minutes) Students identify whether or not the triangles are similar. For pairs of triangles that are similar, students identify the criterion used: AA, SAS, or SSS. Lesson 17: The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 260 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY Exercises 5โ10 5. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement. ๐ ๐. ๐ ๐. ๐๐ โ โ ๐. ๐๐ ๐. ๐๐ ๐. ๐๐ There is no information about the angle measures, so we cannot use AA or SAS to conclude the triangles are similar. Since the side lengths are not proportional, we cannot use SSS to conclude the triangles are similar. Therefore, the triangles shown are not similar. 6. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement. ๐. ๐๐ ๐ = ๐. ๐๐๐ ๐. ๐ Yes, the triangles shown are similar. โณ ๐จ๐ฉ๐ช ~ โณ ๐ซ๐ฌ๐ญ by SAS because ๐โ ๐ฉ = ๐โ ๐ฌ, and the adjacent sides are proportional. 7. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement. Yes, the triangles shown are similar. โณ ๐จ๐ฉ๐ช ~ โณ ๐จ๐ซ๐ฌ by AA because ๐โ ๐จ๐ซ๐ฌ = ๐โ ๐จ๐ฉ๐ช, and both triangles share โ ๐จ. Lesson 17: The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 261 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY 8. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement. ๐ ๐ โ ๐ ๐ There is no information about the angle measures other than the right angle, so we cannot use AA to conclude the triangles are similar. We only have information about two of the three side lengths for each triangle, so we cannot use SSS to conclude they are similar. If the triangles are similar, we would have to use the SAS criterion, and since the side lengths are not proportional, the triangles shown are not similar. (Note that students could also utilize the Pythagorean theorem to determine the lengths of the hypotenuses and then use SSS similarity criterion to answer the question.) 9. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement. ๐. ๐ ๐. ๐๐ ๐๐ = = ๐. ๐ ๐. ๐๐ ๐. ๐ Yes, the triangles are similar. โณ ๐ท๐ธ๐น ~ โณ ๐ฟ๐๐ by SSS. 10. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement. ๐. ๐๐ ๐. ๐๐ = ๐. ๐๐ ๐. ๐๐ Yes, the triangles are similar. โณ ๐จ๐ฉ๐ฌ~ โณ ๐ซ๐ช๐ฌ by SAS because ๐โ ๐จ๐ฌ๐ฉ = ๐โ ๐ซ๐ฌ๐ช (vertical angles are congruent), and the sides adjacent to those angles are proportional. Lesson 17: The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 262 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY Closing (3 minutes) Ask the following three questions to informally assess studentsโ understanding of the similarity criterion for triangles: ๏ง Given only information about the angles of a pair of triangles, how can you determine if the given triangles are similar? ๏บ ๏ง Given only information about one pair of angles for two triangles, how can you determine if the given triangles are similar? ๏บ ๏ง The AA criterion can be used to determine if two triangles are similar. The triangles must have two pairs of corresponding angles that are equal in measure. The SAS criterion can be used to determine if two triangles are similar. The triangles must have one pair of corresponding angles that are equal in measure, and the ratios of the corresponding adjacent sides must be in proportion. Given no information about the angles of a pair of triangles, how can you determine if the given triangles are similar? ๏บ The SSS criterion can be used to determine if two triangles are similar. The triangles must have three pairs of corresponding side lengths in proportion. Exit Ticket (5 minutes) Lesson 17: The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 263 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY Name Date Lesson 17: The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to Be Similar Exit Ticket 1. Given โณ ๐ด๐ต๐ถ and โณ ๐๐ฟ๐ in the diagram below and โ ๐ต โ โ ๐ฟ, determine if the triangles are similar. If so, write a similarity statement, and state the criterion used to support your claim. 2. Given โณ ๐ท๐ธ๐น and โณ ๐ธ๐บ๐น in the diagram below, determine if the triangles are similar. If so, write a similarity statement, and state the criterion used to support your claim. Lesson 17: The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 264 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY Exit Ticket Sample Solutions 1. Given โณ ๐จ๐ฉ๐ช and โณ ๐ด๐ณ๐ต in the diagram below, determine if the triangles are similar. If so, write a similarity statement, and state the criterion used to support your claim. In comparing the ratios of sides between figures, I found that ๐จ๐ฉ ๐ด๐ณ = ๐ช๐ฉ ๐ต๐ณ because the cross-products of the proportion ๐ ๐ = are both ๐๐. We are given that โ ๐ณ โ โ ๐ฉ. Therefore, โณ ๐จ๐ฉ๐ช ~ โณ ๐ด๐ณ๐ต by the SAS criterion for proving similar triangles. 2. ๐ ๐ ๐ ๐ ๐ ๐๐ Given โณ ๐ซ๐ฌ๐ญ and โณ ๐ฌ๐ฎ๐ญ in the diagram below, determine if the triangles are similar. If so, write a similarity statement, and state the criterion used to support your claim. By comparison, if the triangles are in fact similar, then the longest sides of each triangle will correspond, and likewise the shortest sides will correspond. The corresponding sides from each triangle are proportional since ๐ ๐ ๐ ๐ = ๐ ๐ ๐ ๐ = ๐ ๐ ๐ ๐ = ๐ ๐ , so ๐ซ๐ฌ ๐ฌ๐ฎ = ๐ฌ๐ญ ๐ฎ๐ญ = ๐ซ๐ญ ๐ฌ๐ญ . Therefore, by the SSS criterion for showing triangle similarity, โณ ๐ซ๐ฌ๐ญ ~ โณ ๐ฌ๐ฎ๐ญ. Problem Set Sample Solutions 1. For parts (a) through (d) below, state which of the three triangles, if any, are similar and why. a. Triangles ๐ฉ and ๐ช are similar because they share three pairs of corresponding sides that are in the same ratio. ๐ ๐ ๐ ๐ = = = ๐ ๐๐ ๐๐ ๐ Triangles ๐จ and ๐ฉ are not similar because the ratios of their corresponding sides are not in the same ratio. ๐ ๐ โ ๐๐ ๐๐ Further, if triangle ๐จ is not similar to triangle ๐ฉ, then triangle ๐จ is not similar to triangle ๐ช. Lesson 17: The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 265 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY b. Triangles ๐จ and ๐ฉ are not similar because their corresponding sides are not all in the same ratio. Two pairs of corresponding sides are proportional, but the third pair of corresponding sides is not. ๐ ๐ ๐ โ = ๐ ๐. ๐ ๐ Triangles ๐ฉ and ๐ช are not similar because their corresponding sides are not in the same ratio. ๐ ๐ ๐ โ โ ๐ ๐. ๐ ๐ Triangles ๐จ and ๐ช are not similar because their corresponding sides are not in the same ratio. ๐ ๐ ๐ โ โ ๐ ๐ ๐ c. Triangles ๐ฉ and ๐ซ are the only similar triangles because they have the same angle measures. Using the angle sum of a triangle, both triangles ๐ฉ and ๐ซ have angles of ๐๐°, ๐๐°, and ๐๐°. d. Triangles ๐จ and ๐ฉ are similar because they have two pairs of corresponding sides that are in the same ratio, and their included angles are equal measures. Triangle ๐ช cannot be shown to be similar because, even though it has two sides that are the same length as two sides of triangle ๐จ, the ๐๐° angle in triangle ๐ช is not the included angle and, therefore, does not correspond to the ๐๐° angle in triangle ๐จ. Lesson 17: The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 266 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY 2. For each given pair of triangles, determine if the triangles are similar or not, and provide your reasoning. If the triangles are similar, write a similarity statement relating the triangles. a. The triangles are similar because, using the angle sum of a triangle, each triangle has angle measures of ๐๐°, ๐๐°, and ๐๐°. Therefore, โณ ๐จ๐ฉ๐ช ~ โณ ๐ป๐บ๐น. b. The triangles are not similar, because the ratios of corresponding sides are not all in proportion. ๐จ๐ฉ ๐ซ๐ฌ = ๐ฉ๐ช ๐ฌ๐ญ = ๐; however, ๐จ๐ช ๐ซ๐ญ = ๐ ๐ โ ๐. c. โณ ๐จ๐ฉ๐ช ~ โณ ๐ญ๐ซ๐ฌ by the SSS criterion for showing similar triangles because the ratio of all pairs of corresponding sides Lesson 17: ๐จ๐ช ๐ญ๐ฌ = ๐จ๐ฉ ๐ญ๐ซ = ๐ฉ๐ช ๐ซ๐ฌ = ๐. The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 267 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY d. ๐จ๐ฉ ๐ซ๐ฌ = ๐ฉ๐ช ๐ซ๐ญ ๐ = , and included angles ๐ฉ and ๐ซ are both ๐๐° and, therefore, congruent, so โณ ๐จ๐ฉ๐ช ~ โณ ๐ฌ๐ซ๐ญ by ๐ the SAS criterion for showing similar triangles. 3. For each pair of similar triangles below, determine the unknown lengths of the sides labeled with letters. a. The ratios of corresponding sides must be equal, so ๐ ๐ = ๐ ๐ ๐ ๐ ๐ ๐ , giving ๐ = ๐๐ ๐๐. Likewise, ๐ ๐ ๐ ๐ = ๐ ๐ ๐ ๐ ๐ ๐ , giving ๐ = ๐. b. The ratios of corresponding sides must be equal, so Likewise, 4. ๐ ๐ ๐ ๐ = = ๐ ๐ ๐ ๐ ๐ , giving ๐ ๐ = ๐. , giving ๐ = ๐ ๐๐. ฬ ฬ ฬ ฬ intersect at ๐ฌ and ๐จ๐ฉ ฬ ฬ ฬ ฬ , show that โณ ๐จ๐ฉ๐ฌ ~ โณ ๐ซ๐ช๐ฌ. ฬ ฬ ฬ ฬ and ๐ฉ๐ช ฬ ฬ ฬ ฬ โฅ ๐ช๐ซ Given that ๐จ๐ซ โ ๐จ๐ฌ๐ฉ and โ ๐ซ๐ฌ๐ช are vertically opposite angles and, therefore, congruent. โ ๐จ โ โ ๐ซ by alt. int. โ 's, ฬ ฬ ฬ ฬ ๐จ๐ฉ โฅ ฬ ฬ ฬ ฬ ๐ช๐ซ. Therefore, โณ ๐จ๐ฉ๐ฌ ~ โณ ๐ซ๐ช๐ฌ by the AA criterion for showing similar triangles. Lesson 17: The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 268 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY 5. Given ๐ฉ๐ฌ = ๐๐, ๐ฌ๐จ = ๐๐, ๐ฉ๐ซ = ๐, and ๐ซ๐ช = ๐, show that โณ ๐ฉ๐ฌ๐ซ ~ โณ ๐ฉ๐จ๐ช. Both triangles share angle ๐ฉ, and by the reflexive property, โ ๐ฉ โ โ ๐ฉ. ๐ฉ๐จ = ๐ฉ๐ฌ + ๐ฌ๐จ, so ๐ฉ๐จ = ๐๐, and ๐ฉ๐ช = ๐ฉ๐ซ + ๐ซ๐ช, so ๐ฉ๐ช = ๐๐. The ratios of corresponding sides ๐ฉ๐ฌ ๐ฉ๐จ = ๐ฉ๐ซ ๐ฉ๐ช ๐ = . Therefore, ๐ โณ ๐ฉ๐ฌ๐ซ ~ โณ ๐ฉ๐จ๐ช by the SAS criterion for triangle similarity. 6. ฬ ฬ ฬ ฬ and ๐ is on ๐น๐ป ฬ ฬ ฬ ฬ , ๐ฟ๐บ = ๐, ๐ฟ๐ = ๐, ๐บ๐ป = ๐, and ๐๐ป = ๐. Given the diagram below, ๐ฟ is on ๐น๐บ a. Show that โณ ๐น๐ฟ๐ ~ โณ ๐น๐บ๐ป. The diagram shows โ ๐น๐บ๐ป โ โ ๐น๐ฟ๐. Both triangle ๐น๐ฟ๐ and ๐น๐บ๐ป share angle ๐น, and by the reflexive property, โ ๐น โ โ ๐น, so โณ ๐น๐ฟ๐ ~ โณ ๐น๐บ๐ป by the AA criterion show triangle similarity. b. Find ๐น๐ฟ and ๐น๐. Since the triangles are similar, their corresponding sides must be in the same ratio. ๐น๐ฟ ๐น๐ ๐ฟ๐ ๐ = = = ๐น๐บ ๐น๐ป ๐บ๐ป ๐ ๐น๐บ = ๐น๐ฟ + ๐ฟ๐บ, so ๐น๐บ = ๐น๐ฟ + ๐ and ๐น๐ป = ๐น๐ + ๐๐ป, so ๐น๐ป = ๐น๐ + ๐. ๐น๐ฟ ๐ = ๐น๐ฟ + ๐ ๐ ๐(๐น๐ฟ) = ๐(๐น๐ฟ) + ๐๐ ๐น๐ฟ = ๐ 7. ๐น๐ ๐ = ๐น๐ + ๐ ๐ ๐(๐น๐) = ๐(๐น๐) + ๐๐ ๐น๐ = ๐ One triangle has a ๐๐๐° angle, and a second triangle has a ๐๐° angle. Is it possible that the two triangles are similar? Explain why or why not. No The triangles cannot be similar because in the first triangle, the sum of the remaining angles is ๐๐°, which means that it is not possible for the triangle to have a ๐๐° angle. For the triangles to be similar, both triangles would have to have angles measuring ๐๐๐° and ๐๐°, but this is impossible due to the angle sum of a triangle. Lesson 17: The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 269 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M2 GEOMETRY 8. A right triangle has a leg that is ๐๐ ๐๐ฆ, and another right triangle has a leg that is ๐ ๐๐ฆ. Can you tell whether the two triangles are similar? If so, explain why. If not, what other information would be needed to show they are similar? The two triangles may or may not be similar. There is not enough information to make this claim. If the second leg of the first triangle is twice the length of the second leg of the first triangle, then the triangles are similar by SAS criterion for showing similar triangles. 9. Given the diagram below, ๐ฑ๐ฏ = ๐. ๐, ๐ฏ๐ฒ = ๐, and ๐ฒ๐ณ = ๐, is there a pair of similar triangles? If so, write a similarity statement, and explain why. If not, explain your reasoning. โณ ๐ณ๐ฒ๐ฑ ~ โณ ๐ฏ๐ฒ๐ณ by the SAS criterion for showing triangle similarity. Both triangles share โ ๐ฒ, and by the reflexive property, โ ๐ฒ โ โ ๐ฒ. Furthermore, ๐ณ๐ฒ ๐ฑ๐ฒ = ๐ฏ๐ฒ ๐ณ๐ฒ = ๐ ๐ , giving two pairs of corresponding sides in the same ratio and included angles of the same size. Lesson 17: The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M2-TE-1.3.0-08.2015 270 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
